The present invention relates generally to diffractive optical elements, and more specifically the invention pertains to a method to synthesize phase weights for derived diffraction patterns for optical elements in applications such as a phase-only spatial light modulator (SLM).
Any diffraction pattern can be produced in the Fourier plane by specification of a corresponding input plane transparency. Complex-valued transmittance is generally required but, in practice, phase-only transmittance is used. Many design procedures use numerically intensive, constrained optimization. What is needed instead is to introduce a non-iterative procedure that directly translates the desired, but unavailable, complex transparency into an appropriate phase transparency such that at each pixel the value of phase is pseudorandomly selected from a random distribution whose standard deviation is specified by the desired amplitude, and to derive statistical expressions and use them to evaluate the approximation errors between the desired and achieved diffraction patterns.
This invention is motivated by a desire to design phase-only filters and diffractive elements with a small amount of electronic computation and thereby permit programming of arbitrary spatial modulation at real-time rates. Popular design procedures (e.g. the Dammann grating, simulated annealing, iterative constrained optimization, and other iterative procedures) are only practical if performed off-line due both to the numerical cost of performing Fourier transforms repeatedly and the further cost of evaluating the sensitivity of the transform with respect to a large number of pixels (frequently every pixel of the input plane spatial light modulator.) Of course, the solutions can be precomputed and stored in memory, but only if the number of designs required is not too great.
There are many procedures in the area of computer generated holography, esp. kinoforms that permit direct synthesis of the input plane. These presuppose that the Fourier transform pair between the fully complex-valued input and Fourier planes are known and work by encoding the desired complex values to appropriate phase settings. The direct synthesis design procedures thus allow programming at real-time rates, if the desired Fourier plane pattern is known. The amount of memory is also minimized if the complex valued Fourier transform pair can be written as an easily computed function.
The task producing an optical wave with a predetermined function is alleviated, to some extent, by the systems disclosed in the following U.S. Patents, the disclosures of which are incorporated herein by reference:
U.S. Pat. No. 5,258,996 issued to Fraser, et al; PA1 U.S. Pat. No. 5,187,484 issued to Stove; PA1 U.S. Pat. No. 5,184,218 issued to Gerdes; PA1 U.S. Pat. No. 5,142,289 issued to Peterson; PA1 U.S. Pat. No. 5,252,981 issued to Grein, et al; PA1 U.S. Pat. No. 5,012,253 issued to Schuster; PA1 U.S. Pat. No. 4,995,102 issued to Ichinose, et al; and PA1 U.S. Pat. No. 5,276,636 issued to Cohn, et al.
The patent to Ichinose et al discloses a reversed spiral scanning method used by laser radar. The remaining patents are of interest.
Most frequently the direct procedures operate on a small number of adjacent pixels together as a group that approximates several discrete settings over the complex plane (i.e., cell-oriented encoding). This however reduces the space bandwidth, which is already quite small (say 128.times.128 pixels) for current spatial light modulators, as compared to traditional fixed pattern holographic and diffractive optical elements. The procedure of the present invention is also a direct method, but one for which a continuous value of phase is selected for each individual pixel independent of all other settings (i.e., point-oriented encoding.)